Equiform bundle motions in \(E_3\) with spherical trajectories. I
type: Item
instance of: scholarly article
title: Equiform bundle motions in \(E_3\) with spherical trajectories. I
zbMATH Open document ID: 0907.53008
author: Anton Gfrerrer
author: Johann Lang
published in: Beiträge zur Algebra und Geometrie
publication date: 1999-02-18T00:00:00Z
full work available at URL: https://eudml.org/doc/226763
full work available at URL: no.2
review text: In 1906 and 1908 Bricard and Borel published two long papers in which they determined Euclidean motions with spherical trajectories [\textit{R. Bricard}, J. l'École Polytechnique, II. Sér. 11, 1-93 (1906; JFM 37.0705.04); \textit{E. Borel}, Mém. Sav. Étrangers, Paris 33, 1-128 (1908; JFM 39.0749.02)]. Based on these papers the authors determine equiform bundle motions which have spherical trajectories. They introduce a 10-dimensional parameter space \(P(W)\) in which a 6-dimensional surface \(B_6\) represents the condition that a point remains on a sphere during an equiform motion. Furthermore, an embedding of the motion group into a 10-dimensional projective space \(P(V)\) is used to give an interpretation of the sphere condition as an orthogonality relation between the points of \(P(V)\) and \(P(W)\). It is shown that the images of sphere conditions given by any motion are on a linear intersection of \(B_6\). Because of this fact, linear subspaces of \(P(W)\) and their intersection with \(B_6\) are studied. The case that a line is contained in \(B_6\) results in either one point having a circle as trajectory or the points of a circle or a line of the moving system are constrained to move on spheres or in planes. The case of the intersection of a 2-dimensional linear space with \(B_6\) yields a one parameter motion which moves the points of a bundle plane on spherical trajectories. [For Part II, see the following review Zbl 0907.53009].
Mathematics Subject Classification ID: 53A17
zbMATH DE Number: 1208007
zbMATH Keywords: equiform bundle motion
zbMATH Keywords: spherical trajectories
zbMATH Keywords: principle of transference
reviewed by: Manfred Husty
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1208007